
theorem Th4:
  for a being Ordinal holds a *^ a = exp(a,2)
proof
  let a be Ordinal;
  consider fi being Ordinal-Sequence such that
    A1: exp(a,2) = last fi & dom fi = succ 2 & fi.0 = 1 and
    A2: for c being Ordinal st succ c in succ 2
      holds fi.succ c = a *^ (fi.c) and
    for c being Ordinal st c in succ 2 & c <> 0 & c is limit_ordinal
      holds fi.c = lim(fi|c) by ORDINAL2:def 16;
  succ 0 in succ succ 0 & succ succ 0 in succ 2 by ORDINAL1:6;
  then A3: succ 0 in succ 2 & succ succ 0 in succ 2 by ORDINAL1:10;
  exp(a,2) = fi.2 by A1, ORDINAL2:6
    .= a *^ (fi.succ 0) by A2, A3
    .= a *^ (a *^ (fi.0)) by A2, A3
    .= a *^ a by A1, ORDINAL2:39;
  hence thesis;
end;
